Generalized Euclidean stars with equation of state

Abstract

We consider the general case of an accelerating, expanding and shearing model of a radiating relativistic star using Lie symmetries. We obtain the Lie symmetry generators that leave the equation for the junction condition invariant, and find the Lie algebra corresponding to the optimal system of the symmetries. The symmetries in the optimal system allow us to transform the boundary condition to ordinary differential equations. The various cases for which the resulting systems of equations can be solved are identified. For each of these cases the boundary condition is integrated and the gravitational potentials are found explicitly. A particular group invariant solution produces a class of models which contains Euclidean stars as a special case. Our generalized model satisfies a linear equation of state in general. We thus establish a group theoretic basis for our generalized model with an equation of state. By considering a particular example we show that the weak, dominant and strong energy conditions are satisfied.Comment: 15 pages, Submitted for publicatio

Radiating star with a time-dependent Karmarkar condition

“…The Santos junction conditions leads to a differential equation governing the temporal behaviour of the model. Solutions of this junction condition included ad hoc assumptions of the gravitational potentials, applying the method of Lie symmetries and 'spotting' of particular solutions [22][23][24].…”

The effect of a two-fluid atmosphere on relativistic stars

“…Following this, Abebe et al [16], employing the same technique, investigated the behaviour of radiating stars in conformally flat spacetime manifolds. Abebe et al [17,18] used Lie analysis to find radiating Euclidean stars with an equation of state. Collapse models with internal pressure isotropy and vanishing Weyl stresses were also probed by Maharaj and Govender [19].…”

Riccati equations for bounded radiating systems

“…It has already been proved that the method of Lie theory of extended transforms applied to differential equations is very useful for obtaining exact solutions for a general relativistic star. For example, in the previous study, general radiating stars have been studied by Abebe et al [11] and a set of new exact solutions have been found. Recently, Ivanov [13] studied the same problem by introducing a horizon function.…”

“…Exact models with acceleration, expansion and shear were presented by Thirukkanesh et al [6], Thirukkanesh and Govender [7], Herrera and Santos [8], Govender et al [9] and Govinder and Govender [10]. Abebe et al [11] obtained a generalized class of Euclidean stars, with a barotropic equation of state, using the method of Lie symmetries on differential equations. The usual Lie method may be extended to include exponential Lie symmetry generators which produce the most general exact solutions as recently demonstrated by Mohanlal et al [12] Another interesting approach was followed by Ivanov [13] in which he introduced a new variable called the horizon function.…”